@MISC{Ek247measurabilityof, author = {Sm Id Ek and Let N N}, title = {Measurability Of Some Sets Of Borel Measurable Functions On}, year = {247} }

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Abstract

. In the paper we show that the space of injective Borel measurable functions and the space of functions, which norm attains supremum at exactly one point, with supremum metric are coanalyticly hard by using the space of trees. In this paper we show that the set of injective functions is not Suslin in the space of Borel measurable functions f : [0; 1] \Gamma! [0; 1] with the supremum metric. This answers a question of A. H. Stone posed after the problem of [DS], whether the set of injective functions is Borel measurable in the space of Lebesgue measurable functions f : [0; 1] \Gamma! [0; 1] with the supremum metric, was solved by Miroslav Chleb'ik. We say that M is a Polish space if M is a complete separable metric space. Let M be a topological space and P be a metric space. Then B b (M , P ) denotes the space of all bounded Borel measurable functions f : M \Gamma! P with the supremum metric. The space of continuous bounded functions is denoted by C b (M;P ) for P = R it is a normed l...